Everybody pretty much knows you have to have some offbalance before you can do a throw. The Kodokan brainiacs broke it down into eight forms of balance breaking (happo no kuzushi). They basically theorized that it is possible to throw someone's balance out of whack forward, backward, left, right, or in any diagonal.
For simplicity, I break it down into two (maybe I chould call it niho no kuzushi???) - perpendicular and parallel. These correspond to the forward-backward direction (perpendicular to the feet) in the chart below and the direct sideward direction (parallel with the feet) in the chart. Big problem with learning the throws by pushing a static, cooperative uke into offbalance then throwing him is that uke is never static nor cooperative in randori. Bugger always steps somewhere to recover his balance. Well, it turns out that there is a magic moment, when these parallel and perendicular offbalances work best, that is when the moving foot touches the ground. In the instant that the foot hits the ground, a puh or pull in one of these two axes will upset the uke's balance, and will also, as a side effect, tend to slow him down for a moment.
So, in summary, pay attention to the moment that uke's moving foot touches the ground and during that instant, give him a little push or pull (almost just a bump or touch) directly parallel or perpendicular to the line of his feet. See doesn't he freeze for a moment as his body tries to compensate for that offbalance.
There's more to come on the subject of kuzushi. Specifically the classic exercise, happo no kuzushi, and some slight modifications to improve it as a teaching tool.