The 27-year-old arrives at the Prostar Stadium on a free transfer.
Neal started last season at Preston, enjoying a month-long loan spell at Notts County in November, before signing on at Brunton Park back in February until the end of the campaign.
Shrews manager Paul Simpson, who also remains hopeful of Chris Humphrey signing this week, told the club's official website: "Lewis is a good player who has a fantastic left foot and has excellent energy levels with the ability to go past people. He can also deliver good set-pieces.
"Everyone will look at him as a direct replacement for Ben Davies and they are probably not far off the mark. Lewis is a different player to Ben and I am hoping he will give us a real balance to the squad down the left-hand side."
Neal added: "I worked under the gaffer at Preston for around 18 months so I know him quite well. Hopefully he knows what I'm about and I'm looking forward to getting down to the real business."