I don't know how many of you have played Fate, but there's an additional method you can use for Inspiration that can get your players directly involved and also thinking about there characters... which is having them look at their TIBFs at the moment they are trying to make a check, and if any of them directly apply to the situation and check they explain and justify why it applies. Then you can give them Inspiration/Advantage on that check immediately.
So for instance, the party is climbing a high wall and one of the group falls from it. A player beneath says they are trying to grab ahold of the falling character as he goes past, and the DM says to make a STR (Athletics) check. The player looks at his TIBFs and sees that his Bond is "I'll always go back for my friends", justifies that the PC will always be at his best when his friends need rescuing, and the DM agrees and grants Inspiration/Advantage on this upcoming check.
Or another example... a character has a monster tied up and is interrogating it. Negotiation is not going well, so she decides she's going to intimidate it by whipping the monster repeatedly. The DM says okay, make a CHA (Intimidation) check, and the player notices her PC's Flaw is "Takes things just a little too far sometimes". So she points this out to the DM and says her character goes overboard on the beating, the DM agrees, and grants Inspiration/Advantage on the CHA check (with the results of the check being a little bit more than normal due to her character taking things too far.)
In Fate, this is called "Invoking an Aspect" (Aspects being their term for the TIBFs). It gets the players directly involved in acting in accordance to the important facets of their characters. For D&D... it will definitely help players see how their actions can be done in direct relation to their character choices by connecting a check they need to make immediately to what their character is about... rather than try to act "in character" during just general roleplaying situations, the DM needing to remember to give them an Inspiration point for doing so that they're meant to save, and then the player needing to remember to spend it at some other point in the future.