Well, one way to do away with multiclassing barriers and "dipping" is to do away with classes altogether- if the system were more fluid, like HERO, for instance, you could easily build the PC you want...
Of course- that wouldn't quite be D&D, now would it.
OTOH, giving each class a menu of class-specific options available at various levels would make for a great variety of PCs (like what they did with 10+ level Rogues)...especially if they weren't unique to a particular class. Blur the lines, if you will.
For example, the Rogue ability Crippling Strike is pretty cool...but it would work equally well for a Monk, or perhaps a Ranger (for his Favored Enemy only). Perhaps, for those other classes, it would also be a 10th level ability. Perhaps it would be available to other classes at a higher level.
Dipping into spellcasting classes is fairly common. What if the non-spellcasting classes had an ability similar to Extra Spell that would allow them to choose a single spell that they could cast at some minimal level? Your PC doesn't sacrifice his BAB or whatever, but gains a little surprise, like a rogue who can toss a Grease behind him while escaping, or a Fighter who knows Dispel Magic.
Sounds like the Player's Option rules? Yep. Except this time around, the designers would (hopefully) pay a little more attention to balance.