'In the name of Allah the most Gracious and Merciful, It is not necessary to be in a state of wudu to do zikr/tasbeeh. It is, however, recommended to do be in the best of states when doing zikr/tasbeeh.
The Prophet did not reply to salaam without wudu because salaam is the name of Allah.[i]
To get maximum benefits it is best that you are in the state of wudu when doing one of the beautiful acts like zikr.
Sunan Ibn Majah, The Book of Purification and its Sunnah, It was narrated that Abu Hurairah said: A man passed by the Prophet while he was urinating, and greeted him with the Salam, but he did not return the greeting. While he finished, he struck the ground with his palms and did dry ablution (Tayammum), and then he returned the greeting.