LEDs tend to have a characteristic voltage drop, but the thing they care most about is the current through them. Those LM39xx chips have the LEDs wired to power supply on one end, then they bring the output pin on the IC to a logic low to light each one. It won;t care if the Vdrop on the LED is 1.2v or 2.1v.
Here is the data sheet on the IC, and if you just google LM3915, a number of VU meter circuits pop up.
I like to strip old dead gear down to the last screw. I find it good mindless activity when I need a break. I find stereo receivers and old cassette decks often have LED VU meters, and in many cases the whole meter is one little board. SO I save them, probably never use them for anything, but what the heck, a box of VU meters. Point being there is a ready made circuit and display, for free. Usually a four-pin connector: Power, ground, L signal, R signal. Hook it up on the bench, works like a charm.
http://www.ti.com/lit/ds/symlink/lm3915.pdf