Limp caller check min raises on a paired board

[mojed]

Villain is new to the table (why is this always the case?!) so no reads.

Poker Stars, $0.05/$0.10 NL Hold'em Cash Game, 6 Players
LeggoPoker.com - Hand History Converter

BTN: $28.20
SB: $4.90
BB: $10.25
UTG: $50.90
MP: $8.65
Hero (CO): $30.85

Pre-Flop: K[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/diamond.gif[/img] dealt to Hero (CO)
UTG folds, MP calls $0.10, <font color="red">Hero raises to $0.50</font>, 3 folds, MP calls $0.40

Flop: ($1.15) 4[img]/images/graemlins/club.gif[/img] J[img]/images/graemlins/diamond.gif[/img] J[img]/images/graemlins/heart.gif[/img] (2 Players)
MP checks, <font color="red">Hero bets $0.70</font>, <font color="red">MP raises to $1.40</font>, Hero calls $0.70

Turn: ($3.95) 5[img]/images/graemlins/spade.gif[/img] (2 Players)
<font color="red">MP bets $1.70</font>, Hero calls $1.70

River: ($7.35) 3[img]/images/graemlins/heart.gif[/img] (2 Players)
MP checks, Hero???????

I'll explain my line. When getting check min raised, I thought I was WA/WB and would gain little from a reraise. I was way ahead of any underpair, which may have decided this was a good flop to take the pot away from the preflop raisor. I was way behind any J.

So I called hoping to induce an underpair to keep playing at me. Same applies on the turn.

He checks on the river, so given how weakly I've played it I'm putting him on an underpair (or complete air), and figure there's some value to be obtained. Should I bet for value here, how often do I run into a J/full house?

Comments on my whole line would be appreciated too.