My experience:
Beets, never tried.
Spinach won't germinate well in warm soil, same with carrots. Lettuce is the same but seems to tolerate more warmth that spinach. I've also found that spinach seed isn't a great keeper and germination rates fall off significantly with time. This very well may be due to stuff I do or don't do, so try it yourself yrmv.
Temperature generally isn't an issue in the spring for obvious reasons.
For the "fall" plantings, I cover the soil with shade cloth or landscape fabric for a couple or 3 days and water every day. This has a dual purpose, to cool the soil underneath ad little and to evenly distribute moisture in the top few inches.
I soak the spinach seeds for a few hours before planting (put them in a cup of water in the morning, plant them in the evening). When I plant, I pull back the fabric and make several shallow "V" bottomed trenches in the moist soil then sprinkle the seeds in along the trenches very unscientifically spaced (ie tight).
Without covering the seed, I re-cover with the cloth and then water heavily immediately. This seems to cover the seeds just enough and it leaves the early sprouters with a little headroom.
This is the carrot box I seeded a few days ago; carrots take a while to germinate so I'm not worried (
yet.) You can see the "trenches" in both shots.


I also experimented earlier using straw over spinach seed planted using the "broadcast and rake" planting method. I got great initial results almost right away and thought I was on to something. Unfortunately, germination ceased as quickly as it had started. Looking back I'm pretty sure it was due to night time temperatures. We had several low 60s evenings in the week after I put the seed in but temperatures have jumped back up again since (90s in the day, barely under 70 at night). I'm hoping when fall finally arrives to stay I'll have more sprouts.
This is what it looks like as of yesterday; I promise you the seeds were broadcast much more tightly that this and that this is maybe 15-20% germination so far.
