The poetry Duke Orsino speaks to express his thoughts about love is actually merely typical Shakespearean blank verse in iambic pentameter. If we take a look at the ending of each line in his first speech, we see that there actually is no consistent rhyme scheme. Most of the words do not rhyme, so we know he is certainly not speaking in sonnet form here, which is another typical poetry form Shakespeare uses in his plays. Since there is no rhyme scheme, we know that the poetry form here is what is known as blank verse. Blank verse is simply lines written in iambic pentameter that are non-rhymed. Iambic pentameter is a line of poetry 10 syllables long total and containing five iambs. Another word for iamb is foot. A foot contains two syllables, one syllable that is unstressed followed by one syllable that is stressed (Dr. Shwartz, "Shakespearean Verse and Prose"). It is because of the presence of iambic pentameter that we hear this passage as poetry even though it actually contains no rhyme scheme.
The only exception to the blank verse is that an internal rhyme plus a couplet that can be found in the lines:
O, it came o'er my ear like the sweet sound,
That breathes upon a bank of violets,
Stealing and giving odour! Enough, no more;
'Tis not so sweet now as it was before. (I.i.5-8)
In lines 7 and 8 of this passage, the word "odour" rhymes with "more," creating an internal rhyme, which is a rhyme that appears with relation to words in a line or passage, as opposed to with relation to the words at the end or beginning of lines in a passage. In addition, the word "more" rhymes with "before" in the next line, creating a couplet, which is one pair of rhyming words. The use of rhyme with respect to these two lines creates added emphasis. Here he feels a sudden change of emotions from desiring his longing for love to be fed by the music to no longer desiring his emotions to be fed. Hence, using the rhyming words helps to emphasize this sudden, dramatic change in emotions.