Will 1×4 or 1×6 boards span 3 feet between beams for a walking bridge

It has to do more with the quality of the the 2x6s vs the 4x6. A 4x6 x 12 board would have to be clear all the way through, with no cracks or knots. Most softwood logs won't produce this board, and if it cracks, it is likely to break more easily vs the 2x6.

On the other hand, drilling a bunch of holes in a 2x6 does it no favors, though because the two boards are tied together, weak spots in one board are often compensated by its partner.

So, if you could find a really good 4x6, go with it, but otherwise use the 2x6. In most cases, the 4x6 will be more expensive due to its rarity.

In colonial times, they would have gone with the 4x6, as they had lots of hardwood timber and each individual cut took a long time.

The Scout Pioneering website contains information on how to build several bridges. I am guessing that most of these are more complicated than what you had in mind, so if you want something simpler, you can attempt flat span bridge using 2x12's for a span between 14 and 18 feet. On the latter page the author goes on to say that "[i]f you are thinking of building any bridge with a span greater than 20 feet, you better get a structural engineer involved, plain and simple", which I am inclined to agree with.

Note that none of those links say anything about the actual load-bearing capacity of the bridges, but I would assume that most of them would work fine for one or a few people crossing at a time.

If you want to calculate maximum load, it is complicated, especially as the materials in the bridge are likely to deterioriate over time - especially if the environment is wet. Some simple calculators for point loads and distributed loads can be found online to give you an idea of what your bridge can take, but as they make clear on their website, they provide no guarantees. In other words, if you want someone to guarantee you that your bridge is safe, you'll most likely have to hire an engineer, as circumstances change from bridge to bridge.