Villanelle

Laurie... I love what you're doing here but unfortunately, to make it a villanelle you're going to have to shift some lines around, because you've got the repeating idea right but your rhymes are in the wrong place! 

The repeaters have to rhyme with each other.  The second rhyme comes in on the b lines, non-repeating.  So, the first and third line of every stanza should rhyme, and the middle line should rhyme with every other middle line (until the last stanza, where there are 4 lines so no middle... maybe I should just say the second line of every stanza).  It's also very helpful to split the poem up into those stanzas so it's clear that each one ends with a repeating line.  

Like this:

Villanellentine

The magic of a greeting card and rose -
Such sentimental sighings they invoke!
I should not be a cynic, I suppose.

Each year the pile of cardboard refuse grows,
And bouquets send a million suitors broke:
The magic of a greeting card and rose.

Bad poetry will fall to all-time lows,
And clichés will make sobbing women choke -
I should not be a cynic, I suppose.

A sniff will risk a prick in tender nose,
But all trespasses disappear like smoke:
The magic of a greeting card and rose.

Abandonment of common sense just shows
That romance is a sad and sorry joke -
I should not be a cynic, I suppose.

In every bed a man so sweetly throws
His leg across his woman for a poke:
The magic of a greeting card and rose -
I should not be a cynic, I suppose.

Tell you what, though, I like the patterns you've set up in your poem... call it a Laurielle and run with it!