To get wood that doesn't leach tannins, get wood that is TOTALLY dry. Not just dry for a few years but totally dry all the way through!
Some reading on tannin may help with this idea. Tannin is the color that comes from the moisture/sap or whatever we want to call it in wood. It's often found mostly in the layer just under the bark. So it is often a good idea to avoid wood will bark still intact as it has not dried enough to fall off. So we can often avoid the hassle of color if we avoid the sap. As with most things in the hobby, it is not a 100% sure bet as there are some woods which seem to never stop but at least I like to go with the best bet and that is dry wood. So finding how to tell dry from somewhat dry is a starter.
Dry wood may have a musty, natural smell but not anything that smells like a Christmas tree! The conifers like cedar, fir, spruce hold their sap and often smell so avoid that smell. Go for wood that feels lighter than it might if wet. Hard to judge if not used to working with that wood but good if you can find it. Look at the color to compare the inside color to the outer layers. Wood dries from out to inside and as it does the color often changes so if we cut off an end of a piece we can see when the outside has dried but not the inside. Often the dry will be a lighter color, so going for a white wood has a better chance than a dark wood.
Put all these points together and you have a good shot at missing all the trauma of tannins!!