This is a fast path which preserves the character of the for the top but sacrifices the bottom surface of the table (the side seen by the floor). Assuming you have a planer or are savvy with hand planes, this is approachable without too much pain. Find the thinnest board in the set you wish to use and plane the bottom surface flat.

For the sake of discussion, lets assume this is now 1" thick and flat on one side. Now plane the rest of the boards (on one side only), to the same 1" thickness. Now you have a set of 1" thick boards which have "character" on all faces except one. The can rest flush on your stretchers and still look interesting. You can stop here and build the table as you wish. If the fresh wood underneath seems to bright a color, a little stain will help greatly to bunt that. Note: it is worth trying the boards in different combinations, to see if one combination is more appealing than another.

If there is too much character and the gaps between the planks bothers you, you can plane the side surfaces, however, it is probable that fresh wood will be visible to folks sitting at the table. In a perfect world, it wouldn't but this kind of woodwork is far from perfect. If all the planks are planed on their edges as well this can be minimized. But a close inspection will still see fresh wood. A little dark stain goes a long way, so don't over do it. You are looking for a color just a bit darker than the surface. Some folks will 'distress' the edges as well to help blend the overall look. (This, too, is easy to overdo).