There are a number of ways.
Use a jointer to straighten one edge, bandsaw and then use a jointer, and fasten a straight board to run against the fence.
You can fasten the straight board with clamps, or use some nails.
If you have sap wood you can nail into that and then cut further in.

In industry there are special saws that power feed straight without an edge reference.

The machine typically used is called a jointer, thus the name "Joint'R" for a gadget meant as a substitute. Unless you want to spend several hundred dollars on a new piece of hardware, you're stuck with improvised measures such as the one you've tried. Hopefully you've got a decent, well-adjusted saw with a sharp blade and a solid rip fence, and a helper to assist in guiding the boards through. Doing without any of those would make this job a chore.

You can clamp a peice of plywood or a straight peice of lumber to the board for a guide then use a circular saw for the cut.

Those suggestions get you a reasonably straight edge, but you also need to make the boards flat. You should have made one face flat before planing to final width. Typically you would use a jointer or handplane to flatten one face, then put the flat face down when running the board through the planer. There's various sleds and jigs you can rig up to sort of flatten/face-joint with the planer, too.

One jig for your table saw has been described in many books, like "Tage Frid Teaches Woodworking", I think it's titled. Depending on how long your boards are, you build a sled long enough to support them, with a stop at the front, and sandpaper on the top at regular intervals. The rough board rides on top of the sled, held at the front by a pin, and at the back by hand pressure holding it down against the sandpaper. You should probably see a picture of this before you try it... otherwise, the circular saw is probably your best bet.

There are different ways, though not necessarily better ways(read, quicker). They will all take time.
Ron

whether you choose any of these ways or not, IMHO the Tage Frid book would be an advantageous purchase.

I think there's a series of cabinetmaking/woodworking books by Mr. Frid. I have the one on joinery somewhere among my clutter. It says #3 on it or somesuch, leading me to believe there's a 1 and 2. Maybe they've been reissued in one volume...sweet.
Casey