A lot of this post will be rephrasing what I've said before, so I probably shouldn't post... but then, this thread is long enough that that is allowed, right?
Most (not all) of my problems with 4e MCing is the name (and what is in a name, really?). It is counter-intuitive to me that a process called multi-classing doesn't give you the aspects of the class you are multi-ing into. The bog-standard fighter/rogue, if started as a rogue, loses (by losing feats, not significant) toughness rather than improving on one of the defining characteristics of the class he is multi-ing into. Now, JohnSnow might say that if he did improve his hitpoints (like a 3e MC), it would be unbalanced because he didn't give up anything significant. While true, that is, to my mind an indictment of the process: 4e MCing *does not* let you "average" your two classes. You don't give up anything key to your class-role, and hence can't pick up anything important to your MCed class-role. How can you call that multi-classing?
I also expect that MC will come to be viewed as weak. Swapping equivalently powered Powers at the cost of a feat might be worth it for increased versatility. However, in 4e different classes use different weapon/implement sets. If you want to multi from Fighter into rogue, you *can't* make use of your sneak attack 1/encounter without either using a gimped weapon (thereby negating any advantage of sneak attacking) unless you spend an extra feat on SAing with a real weapon (which IIRC is possible?). That makes the initial feat +7 damage/encounter at the cost of 2 feats. With longer encounters, that is probably about .35 damage/feat/encounter, less if you have to declare the SA before attacking. Anyone remember if the other rogue Exploits are light-blade dependent?
Or rangers: melee rangers are dual-wielders. Multi-ing into ranger, you aren't getting the at-will dual wield mechanics for free, so when you want to use your cool MCed per/encounter ability, you need to draw an extra weapon, possibly sheath your (non-light) main weapon and, having spent time doing that, lost any advantage it might have earned you. Maybe you can spend feats on dual-wielding as a non-ranger, but then you have to add that feat cost to the MC feat cost.
A fighter that wants to cast fireball has to worry about spending a feat to use his sword as an implement (possible, IIRC), or maybe a feat for quickdraw...
Because (many) classes are distinct in their "weapon" of choice (paladins/fighters/warlords stand a good chance at using the same gear, warlock/wizard/cleric might be), and casters gained implements that serve the same purpose, you need to add in the logistical cost of swapping between main-hand gear sets/getting proficiency in MCed gear sets. If that cost exists, MCing is going to be somewhat gimped.