Rolling initiative every round, to my knowledge, is not really 'done' in d20 versions of D&D, like dnd-3e, dnd-3.5e, and pathfinder. My experience of it is from AD&D 2e.

In previous editions of Dungeons and Dragons, like adnd-1e and adnd-2e, initiative was rolled every round (Advanced Dungeons & Dragons 2nd Edition Player's Handbook, 93), which were each supposed to be roughly a minute long (119). The variability seemed to be intentional, apparently to stimulate the unpredictability and chaos that occurs during battle.

The default system was actually to have each side in a battle roll initiative, then the winning side would take their turn first. Modifiers existed to improve or degrade initiative for each side, and if the initiative came out the same, all actions happened simultaneously (94).

Two optional, more complicated, initiative procedures were available as well: Group Initiative; one roll, but with individual modifiers like weapon speed (94), and Individual Initiative; individually modified rolls for each creature.

If a creature had multiple attacks from multiple weapons, those attacks happened at once on its initiative. For a creature with multiple attacks from high levels, only its first attack happened on its initiative count; all creatures with multiple attacks acted again after the first set of actions in a round, in the same initiative order.

Spells had a casting time, usually a number which modified the initiative of the caster. Spells requiring one round to cast instead took effect at the end of the current round after all other actions. Some spells took 10 minutes (10 rounds) to cast, and taking damage, or failing a saving throw would (not could, would) interrupt the spell (85).

The effects of rolling initiative every round were simply unpredictability; especially for spellcasters, though it tended to balance out. As another answer mentions, holding initiative is an option that might improve predictability, but take care not to go too slow; going first sometimes means that your opposition can no longer take any actions.

It used to be normal

Rolling initiative every round, to my knowledge, is uncommon in d20 versions of D&D, like dnd-3e, dnd-3.5e, and pathfinder. My experience of it is from AD&D 2e. I'd recommend that you look to what was done in those editions that natively used per-round initiative for inspiration.

What's gone before

In previous editions of Dungeons and Dragons, like adnd-1e and adnd-2e, initiative was rolled every round (Advanced Dungeons & Dragons 2nd Edition Player's Handbook, 93), which were each supposed to be roughly a minute long (119). The variability seemed to be intentional, apparently to simulate the unpredictability and chaos that occurs during battle.

The default system was actually to have each side in a battle roll initiative, then the winning side would take their turn first. Modifiers existed to improve or degrade initiative for each side, and if the initiative came out the same, all actions happened simultaneously (94).

Two optional, more complicated, initiative procedures were available as well: Group Initiative; one roll, but with individual modifiers like weapon speed (94), and Individual Initiative; individually modified rolls for each creature.

I'd recommend this old rule to limit the 'swing':

If a creature had multiple attacks from multiple weapons, those attacks happened at once on its initiative. For a creature with multiple attacks from high levels, only its first attack happened on its initiative count; all creatures with multiple attacks acted again after the first set of actions in a round, in the same initiative order.

This old rule is hard to make fit

Spells had a casting time, usually a number which modified the initiative of the caster. Spells requiring one round to cast instead took effect at the end of the current round after all other actions. Some spells took 10 minutes (10 rounds) to cast, and taking damage, or failing a saving throw would (not could, would) interrupt the spell (85).

The d20 ruleset really doesn't look like it would fit that old rule very well. It's one of the things that when changed with the 3rd edition, intentionally or not, factored in to increase casters' power.

In Summary

The effect of rolling initiative every round was largely unpredictability; especially for spellcasters, though it tended to balance out. Consider embracing the unpredictability as increasing the simulation of verisimilitude. As another answer mentions, holding initiative is an option that might improve predictability.

Take care not to go too slow though; going first sometimes means that your opposition can no longer take any actions.