Will 1x4 or 1x6 boards span 3 feet between beams for a walking bridge?
The boards will flex something awful, which makes for an unpleasant walking experience. Tying them together with an unsupported stringer in the middle will make things quite a bit less wobbly. If the bridge is short then an X-shaped brace between the beams will give you much better support with the same look.

I personally would opt for the 1x4 if all other things were equal. Three feet is a really wide span for rotten or insect-damaged boards. Falling through a floor all the way to your hip is hilarious for everyone who sees it, but not an experience I'd be keen to repeat. With a 4" board any way you stand you're spanning two of them.