Here's what rhymes with routu.
to do two too due du dew tu chew deux bu ju jus peu pu boo chou chu doo tue pooh poo who you new true knew blue drew u blew coup qu shoe zu lieu shew su sue vous fu lu rue stew zoo adieu ado brew hu ku loo mu nu pew seu shu sou vu woo yew yu coo cou ewe feu gu outdo ru shou thu tzu vou wou wu goo moo shoo gnu roux thew whew booboo into through few view grew que crew pursue threw flew q anew clue cue glue hue queue undue cu flue slew taboo undo flu piu tattoo thru beaucoup eschew fue hew mieux shrew sprue vnto vue clew mew redo strew spew zebu slue phew achoo review residue screw bamboo canoe renew accrue debut ensue subdue untrue jusqu overdue skew revue shampoo imbue outgrew overdo halloo valuer endue indue kazoo muumuu wazoo regrew atishoo revenue hitherto withdrew avenue rendezvous retinue caribou sinew askew ballyhoo cockatoo honeydew bugaboo fondue foreknew marabou peekaboo bestrew buckaroo overshoe prevue superglue overdrew interview overview construe overthrew kangaroo overflew overgrew barbecue hullabaloo barbeque misconstrue