I don't think the following are things that are specific to 4e or that 4e taught me, but I think they are things that 4e reinforced in my DM philosophy.
Say yes (DMG p. 28) - the basic idea of "if the player can imagine it, he can attempt it" was something that I always had in my games. 4e actually spent time reinforcing it with the "say yes" mentality. This has kept me always looking for that creative spark from the players. I'm more consciously aware that my first response should be to say yes, and then adjudicate how they can attempt it. It's not a matter of fact that they will automatically fail, as the "say no" response usually creates. This has been greatly liberating in my games on both sides of the screen. As a matter of fact when I teach D&D to new players I usually start with the "if you can imagine it, you can attempt it" phrase. That is the huge difference between a TTRPG and a CRPG and it is fantastic to see players going all out in doing "out of the box" things.
Pacing (DMG p. 24) - this is something that I always had in my games but I was quite a bit lax with it. At times this resulted in entire sessions where the players meandered about aimlessly, or "scenes" simply took too long with no purpose. I had my players running around trying to "Find the Fun." Those were the worse sessions during the game, and also in retrospect when I asked the players how they liked specifics of the adventure. 4e spent a lot of time talking to the DM about this. Exploration is perfectly fine and encouraged, but it must have a purpose and payoff. The pace of the game must feel like it's moving. Running around with no information is a waste of time. This lead to the second side of this below.
The Information Imperative (DMG p. 26)
If there's information the PCs absolutely must have in order to continue the adventure, give it to them.
Too often I had been hinging important parts of the adventure on the result of a single die roll or the interaction with a particular NPC/monster. If a player needs to find the journal that has the BBEG plan in exquisite detail, give it to them. What good does it do to hide it inside the most inconsequential nook in a hallway and then get all flustered when the players don't find it? If it's important information then find a way to give it to them. This was also very important for those investigation adventures because I started changing the parameters to move the situation forward. An investigation adventure that constantly depends on the "specific" circumstances to give out information is the most frustrating adventure ever. It becomes a game of "guess what the DM" wants us to do, instead of a fun romp exploring and finding clues. As said before this is not something specific to 4e but 4e does a good job of reminding the DM to pay mind to this.
Know your players - Player Motivations (DMG p. 8) - This is one of those that I've always kept in mind, but 4e really hammered home. During the campaign I was running I decided to use a pre-canned adventure at one point. What I found was boredom. Not because the adventure was bad, but because it did not scratch the itch that the players had. It was not "motivating" to them so it was a series of combats with barely any connection to the PCs. I learned my lesson and the next session I looked for ways to pike the interest of each player based on the things they like to do. Mind you this was a continuation of the same pre-canned adventure. But it became a more ad-hoc experience tied to each PC. The players got into it because they were invested in their own goals. This worked much better and it's something that I now keep in mind at all times. When I'm preparing an encounter I look at how it would fit with the motivations of each player. Not every encounter has something to motivate each player, but every adventure definitely has multiple encounters that will "hook" to each of their motivations.
Improvisation - (DMG p.26) Because of the "say yes" directive I've become much more aware of improvisation. The most important "rule" of improvisation is be prepared. That seems to make little sense, but I find that the more prepared I am for the players the more "realistic" the improvisation becomes when I have to "step off script". My players have been having a wonderful time simply exploring things, and I've had a wonderful time discovering some of these things at the same time with them.
Keep it simple - (Monster Design) - I have had a blast designing new monsters, or altering old favorites to keep players on their toes. Designing Count Strahd Von Zarovich for use during the exploration of Castle Ravenloft was one of the most interesting and rewarding experiences. That combat was awesome. I used the inspiration from P.N. Elrod's
I Strahd as the basis for the monster, then I looked at his stats in the classic I6 adventure and started from there. Here was a creature that was a warrior, a wizard, and a vampire all wrapped up in one. The most interesting part of all of this was the "feel" of the monster as we played. He felt like a warrior, a wizard and a vampire all wrapped up in one. Everything I needed was right there in the stat block. Not once did I have to break the immersion to go look up a rule, a spell, or a piece of equipment. The encounter was fast paced and exciting. That was an extremely fun experience. 4e is tops in the area of monster design IMO, of any edition of D&D. It is simple, focused, succinct, fun, flavorful, and it is totally relevant to the use by the DM where it counts, at the game table.
Exciting combats - Once again not something new with 4e but something that is reinforced in 4e. I'm using terrain, tricks, traps, monsters and exciting locales for combats and that works wonderfully in any game. 4e just made it more fun for me because it doesn't require a lot of "making it up." The underlying framework presented gives me many examples of how things can work, and I can use those or make my own derived from the framework.
Reskinning - This is something I found fascinating working with the framework for 4e. If I wanted to play X type of character, and there was no existing class, I could rather easily reskin an existing one to fit that concept. This was refreshing because it reminded me of my BD&D and AD&D games where players would come up with alternate class concepts based on the existing ones, like the swashbuckler, etc. I also found the balance within the game rules refreshing because as a DM I could more easily help my players if they wanted to "change" something without becoming overpowering. Themes were a breath of fresh air in the design space for this purpose. The underlying mechanics did not have to break because the flavor was different. The same with powers, which have seen a lot of reskinning in my games. If a player wanted a power to work more as a mental power than a weapon power we could make that happen without breaking anything. This was liberating in a sense because the fear of breaking something was gone.
Expanding the design space - I found that the solid 4e framework allowed me to expand the design space with things that the game didn't have but I wanted in certain circumstances. These were things like Injuries, Wounds, Diseases, Curses (Hexes), flavorful magic items, etc. Designing was easier because I had a solid footing that already worked.
Running higher level games - For the first time since I'd been playing D&D I found a way to have high level games that did not break, and did not suck to prepare. I was able to keep players challenged and we all had fun.
I'm sure that I'll remember more as I go along.