My game is a very high-power, high-color setting with an extensive background and excellent reason for the existence of all the 'weird' races like Tabaxi (that being the apocalyptic event that caused the downfall of the Golden Age empire, leaving mutating radiation behind a la Gamma World). So not only do I not mind +LA races, I positively encourage them; in fact, I actually modified the core races for my game such that they all get extra benefits and powers above the PHB stats. I estimate that each of them is worth about +2.
Interesting thing is, since all the races have that +2, there's no reason to actually use it in game (the point of LA is to balance powerful PCs against less powerful ones). So all the races of my game have 2 less LA than they would anywhere else. Thus, a core-rules Drow would be allowed in my game sans LA at 1st level, if I used core rules Drow (they're actually modified in this setting too). None of the races have racial HD, even if they would in core rules; thus, the race of Minotaurs in my game is actually considerably less powerful than that in the Monster Manual, starting with only 1 hit die like everybody else and only +6 STR, +2 CON and -2 CHA for their stat adjustments. This sort of fiddling has allowed me to throw in wildly different creatures like PC illithids (though the illithid subraces all have a LA +1 or higher even with the subtracted-out 2) and thri-kreen (no LA). Encounter-wise, I just keep the phantom +2 levels in mind, and don't pull punches when throwing more powerful monsters at my PCs.
We also use the UA buyoff rules, so most of the oddball PCs in my games, past and present, have bought theirs off by the time the game gets into high-level territory. The sole exception in presently-running games is one of the PC dragons in the Epic party- she never bought her LA off and is still playing with +3. But since the party's now 28th, and she
is a dragon of age category 8 or so even with the LA, nobody really minds much.
Even with this vast wealth of options though, most players have preferred to stick with basic humans and elves, or at any rate human subraces and elf subraces. I suppose they find it easier to make a 'roleplay connection' with creatures whose lives are not dissimilar to their own. That said, my two currently-running games do feature a few oddballs. The Epic game has two PC dragons (one a homebrew race called a Pink dragon, who never bought off her LA, the other a Deep dragon who did buy his off), a plains-dwelling lion-aspected Tabaxi variant called a Leonal, and a templated (half-Water Elemental) elf.
The other game (presently at 16th level) features a Psylon (one of the illithid subraces) Telepath, the nominal party leader, and until recently featured a Grippli Cleric. The frog's player was dissatisfied with how his character was working out in the game (sadly the Telepath was partly responsible for this, since he figured out a way within the psionics rules to do the Cleric's main schtick better than he did), so he requested a new character. That new character joined last session, and is a Crystalloid (living crystal creature sort of like a crystalline Earth Elemental, based somewhat on the crystal race from the Robotech Sentinels series) Warlock. This character, at least, fills a unique niche in the party that nobody else is even close to matching, so the problem shouldn't recur.
The Crystalloid bought off his LA during character creation- we do that by just subtracting out the XP that would have been paid to do it, from the character's starting total (whatever that turns out to be). If that subtraction ends up reducing the character's final level, then oh well; the level gets reduced. The character will make it up by gaining more XP over time, we figure- it seems to work out fairly well in practice.