In a statement on the CEPU website, Ed Husic called on Thodey to provide certainty and security for its workforce by entering into a collective bargaining agreement "in good faith" with the Telstra unions and their members.
He also called on Thodey to review the business risk presented to Telstra by job cuts made under the telco's transformation and to focus on protecting in-house skills to reposition as a viable NBN participant.
"We congratulate Mr Thodey on his appointment and I'm sure he's aware of the huge range of challenges already confronting him," Husic said.
"The CEPU is ready to work with Mr Thodey - we hope he's ready to do likewise with the CEPU and Telstra employees, so together we can deliver the services Australian consumers deserve."
CEPU is involved in a long-running stand-off with Telstra over its unwillingness to negotiate collective agreements for employees.
The stand-off has resulted in industrial action on a number of occasions.
Husic said Trujillo had damaged Telstra and "left it in a mess".
"Riding a wrecking ball into stakeholder relationships has weakened Telstra - and the times now call for a smarter, inclusive approach," Husic said.
"It will be interesting to see if Thodey remains a creature of the prevailing senior management mindset and culture - or if he will make his own mark."