Hey, Ron W is the real oracle of the electrons around here! But I'll kick in my two cents.
It's a slightly less green, but I'd probably stick with the alkaline batteries myself. I was going to mention this in another thread about manufacturing instruments from recycled/reclaimed materials - it's good to be more earth conscious, but if you're going to consume a battery or a piece of rosewood, I can't think of a more responsible way to honor it by using it in your Alembic. To me, it would be a shame to cut down an old growth redwood to build a deck in your backyard, but if you lovingly built and instrument with a top from that same tree, I think that's a good use of the material.
The technical side of the battery issue is sort of a mix. Adriaan's comment is correct here - the voltage level from the battery is what determines the performance in an audio circuit, not the current or power available in the battery. When you use a 9V battery to power your electronics, it basically splits that power into +4.5V and -4.5V. Those are the rails or extreme levels that the amplifier can operate between cleanly. If you try to exceed the rails, the signal will get clipped off which will start out sounding less dynamic or more compressed and will get worse until it sounds fuzzy and distorted. As the voltage from the battery drops, the rail voltages drop - when the battery gets to 8V, it will be +/- 4.0V, etc. A new single alkaline battery (like an AA battery) is 1.5V; inside the case a 9V battery is a stack of 6 of these, so it produces a total of 9V. A single NiMH cell is typically more like 1.2V, so there are 7 cells in a NiMH 9V battery, ending up at around 8.4V. So you're kind of starting out with a bit of a disadvantage from a headroom standpoint (I think that you can buy NiMH with 8 cells which would actually be a little higher than 9V).
Each battery has different characteristics for voltage, the amount of power they can hold/produce, and how the power degrades over time. Alkalines hold a lot of power and degrade slowly over time. NiMH can withstand a lot of power drain, but when they reach their threshold, it drops off quickly. That means that you may not get that much warning when the rechargeable battery starts to fail.
Finally, if you keep a spare alkaline in your case it should be good for years. If you charge up a NiMH, it will tend to self-discharge on it's own, probably in a month or less, so you need to be more vigilant in keeping things charged up.
Of course, there's nothing wrong with keeping an alkaline in your case as the backup, too.
When you have a device like a digital camera or walkie talkie that would really eat a set of batteries in a hour of use, then it makes sense to look more closely at rechargeables. When you can regularly get hundreds of hours or more of use between battery changes, I think the hassle factor of rechargables is probably not favorable.
David Fung