Short answer: while a flash will work okay with alkaline batteries, other types are better.

Nikon produces excellent manuals for their flashes, with a lot of great technical information (rivaled in this area only by Metz). The following is from the SB-600 User Manual, page 19:

Alkaline-manganese | 3.5 sec. | 200  / 6-30 sec.
Lithium            | 4.0 sec. | 400  / 7.5-30 sec.
Nickel             | 2.5 sec. | 180  / 6-30 sec.
NiCd (1000 mAh)    | 2.9 sec. |  90  / 4-30 sec.
Ni-MH (2000 mA)    | 2.5 sec. | 220  / 4-30 sec.

The first column is the type, followed by approximate time between full-power flashes (best case), followed by the number of flashes you can expect to get, followed by the typical recharge time.

Additionally, on a further chart, Nikon recommends recharging your NiCds or NiMHs when the recycling time drops to 10 seconds, whereas it's suggested to keep your alkalines until they're taking a full half a minute between flashes. (It's worth noting that they give the 10-seconds-then-trash recommendation for Lithium primary cells too.)

So, you can see that while Lithiums last longer, it's not so much more than NiMH that it's really going to be worth it except in extraordinary circumstances (like, no access to power for weeks).

The most important differentiator here, though, is the recycle time. It takes 50% more time to be ready to go again when using alkaline batteries. That alone is a reason to go with NiMH.

Also, if you're not going through batteries constantly, I highly recommend low-self-discharge NiMH batteries like Sanyo's Eneloops — normally, NiMH batteries drain significantly just sitting overnight, and these don't, so you can charge them up and have them ready to go.