Unfortunately, when you get down to 1x3 stock, it is very difficult to find straight pieces. If you're lucky, you can find a couple of long pieces and then cut shorter pieces from the bent stock to meet your needs. If you can't find any long, straight pieces, what are the alternatives.
1) Check for 1x material in the trim area. A lot of times this stock is finger joined and while it is often a softer wood, you can add cross bracing to firm up the frame. You'ld have to do some cross bracing with poplar, so I don't see this as an issue.
2) Why are you set on 1x3 stock
Use 1x4 or 1x5 material. This would be better in the long run anyway. 1x3 is pretty flimsy. If you're really set on 1x3, the get access to a table saw and rip the larger board down.
3) Buy some MDF and have the home center cut the straight pieces you need. The biggest problem here is making a strong joint. Use the right screws and glue and glue blocks behind the joints. The home centers often advertise that they will only make one or two cuts free, but I've never been charged for multiple cuts. I often have them make some cuts just to make transporting home easier. Even if they do charge ($.50/cut), its cheaper than buying a table saw. Again, the MDF would need a lot of bracing to maintain shape, but you're going to do this anyway, right
I'm spoiled now by a well equiped shop, but years ago I rebuilt a kitchen with only hand tools and a mitre saw. It just takes a little bit of imagination to work around these issues.