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The Heroism spell says(PHB, 250):

Until the spell ends, the creature is immune to being frightened and gains temporary hit points equal to your spellcasting ability modifier at the start of each of its turns. When the spell ends, the target loses any remaining temporary hit points from this spell.

So, does that mean that after two rounds and not taking damage the "hero" will now have +6 HP if my spellcasting ability modifier is +3? Or does it just keep refilling to +3?

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No, they do not accumulate.

When granted new temporary hit points, you can choose to take the new "stack" or the old one.

Healing can't restore temporary hit points, and they can't be added together. If you have temporary hit points and receive more of them, you decide whether to keep the ones you have or to gain the new ones. For example, if a spell grants you 12 temporary hit points when you already have 10, you can have 12 or 10, not 22. — PHB, p.198

In your example, the hero will keep refilling their temporary HP to +3.

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  • 2
    \$\begingroup\$ Crawford addresses this nicely in the podcast about temporary hitpoints posted on Feb 2, 2017. \$\endgroup\$ – Slagmoth Jun 14 '17 at 12:35
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    \$\begingroup\$ Does any body have a link to that? \$\endgroup\$ – Simon F Jun 14 '17 at 16:13

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