The procedure is longly decribed in an article of the Azure documentation: https://docs.microsoft.com/en-us/azure/aks/ssh. It consists of running a pod that you use as a relay to ssh into the nodes, and it works perfectly fine:

You probably have specified the ssh username and public key during the cluster creation. If not, you have to configure your node to accept them as the ssh credentials:

$ az vm user update \
  --resource-group MC_myResourceGroup_myAKSCluster_region \
  --name node-name \
  --username theusername \
  --ssh-key-value ~/.ssh/id_rsa.pub

To find your nodes names:

az vm list --resource-group MC_myResourceGroup_myAKSCluster_region -o table

When done, run a pod on your cluster with an ssh client inside, this is the pod you will use to ssh to your nodes:

kubectl run -it --rm my-ssh-pod --image=debian
# install ssh components, as their is none in the Debian image
apt-get update && apt-get install openssh-client -y

On your workstation, get the name of the pod you just created:

$ kubectl get pods

Add your private key into the pod:

$ kubectl cp ~/.ssh/id_rsa pod-name:/id_rsa

Then, in the pod, connect via ssh to one of your node:

ssh -i /id_rsa theusername@10.240.0.4

(to find the nodes IPs, on your workstation):

az vm list-ip-addresses --resource-group MC_myAKSCluster_myAKSCluster_region -o table