Leach has been in the NFL for nine seasons and made the Pro Bowl each of the last three, once with Houston and twice with the Baltimore Ravens.
Financial considerations, more than anything else, led to Baltimore to releasing Leach this week and he has a history with Coach Joe Philbin after playing for Green Bay his first three seasons in the NFL.
Philbin was asked Wednesday what he remembered most about Leach from their time together and replied: "I remember on his first day I had to point him to the practice field and give him a ride to the practice field. That's a true story; that's what I remember about him (laughing). Lost. Totally lost."
Philbin later said he remembered Leach for his toughness.
No question, Leach is a bit of a one-dimensional player, but there aren't many better blocking fullbacks in the NFL.
Evan Rodriguez, on the other hand, is known more as a pass-catching fullback. He's actually a hybrid type of player, a fullback who can play H-back and also line up at tight end.
In that vein, he's very much like Charles Clay, who had himself a very difficult day at minicamp on Wednesday.
On that same day, Rodriguez made his debut in a Dolphins uniform after being claimed off waivers from the Chicago Bears.
A fourth-round pick in 2012 out of UMass, Rodriguez was cut by the Bears for no reason other than a pair of arrests in the offseason.
No question he'll have a short leash with the Dolphins.
"I feel like I'm getting a second chance," Rodriguez said Wednesday. "I'm thankful to the organization for it. That's definitely all behind me. That stuff's in the past. It's a lesson learned. Here's a new opportunity to prove myself."
Rodriguez clearly has ability, but then again so does Jorvorskie Lane, who came out of nowhere to become a starter last year and had his share of key moments.
Lane is a bit of mix between Leach and Rodriguez in that he's a better blocker than Rodriguez and a better offensive player with the ball than Leach.
In addition to those three guys, the Dolphins also invited former Minnesota Vikings fullback Ryan D'Imperio to this week's minicamp.
While he's a long shot, it's just another alternative for the Dolphins.
In the long run, who the Dolphins have at fullback in 2013 may come down to what quality they value most at the position.
Philbin was asked that specific question on Wednesday and replied by saying he was hoping to find someone who could do everything well. It was a way of not tipping his hand.
Who the Dolphins end up keeping at fullback might reveal his true feelings more accurately.