I also have had my first cucumber plant and I must say, I started off with no trellis and the plant suffered from disease (perhaps from being too close to the ground). Eventually I made a DIY trellis to bring the vine and flowers about a foot off of the ground and the results afterwards were encouraging.
I would suggest implementing one into any vine'ish plants.
"Don't be afraid to go out on a limb. That's where the fruit is."
-H. Jackson Browne