The news quickly got better. Fletcher went next.
"I see Shonn's name go up," Fletcher said. "I was just happy for him to get to go to the Jets. Then, to hear mine called next, it was a great feeling. It was good for Iowa football. It was a good deal."
After New York took Greene as the first pick of the third round, the first selection on Sunday, St. Louis stepped up and tabbed Fletcher, the 66th player taken overall.
"I spoke with St. Louis before at the combine," he said. "We had a good conversation. Things went well and I'm happy to have an opportunity there."
Fletcher, who intercepted three passes and broke up 10 others as a senior, spoke with several Rams coaches on Sunday. They told him that he could come in and compete for playing time at cornerback.
"I'm just going to go in and compete and we'll see what happens from there," Fletcher said. "I know they have good players there."
Fletcher was prepared for a long Sunday waiting to hear his name called, but was relieved to get it over with quickly.
"I was going into it with an open mind," he said. "To hear my name called as early as it was, it was a dream come true. I look forward to being able to get on the field here soon."
The Rams are scheduled to hold their rookie camp beginning on Thursday.
"He's a big corner who really came on this year," ESPN Draft Analyst Mel Kiper Jr. said on Sunday's broadcast. "He was one of the hot guys through the draft process. He's real strong in the upper body, really good athlete. You look at this kid, as he progressed, he needs a little more experience. But he closes quickly, athletic, made some plays in the run game as well. He will tackle.
"This was a guy who moved up. He was ascending over the last two months. And there you go, Kirk Ferentz, respected again. His opinions mean a lot to the NFL. You've got back-to-back Hawkeyes at the top of the third round."
The Rams' official website lists three players as first-team corners and three more as backups. Four of those guys come into camp having been drafted in the first three rounds of the last five drafts.