diff --git a/graphene/relay/tests/test_mutation.py b/graphene/relay/tests/test_mutation.py
index b58a3ddf..96ddcf0c 100644
--- a/graphene/relay/tests/test_mutation.py
+++ b/graphene/relay/tests/test_mutation.py
@@ -1,5 +1,4 @@
-import pytest
-from promise import Promise
+from pytest import mark, raises
 
 from ...types import (
     ID,
@@ -55,15 +54,15 @@ class SaySomethingFixed(ClientIDMutation):
         return FixedSaySomething(phrase=str(what))
 
 
-class SaySomethingPromise(ClientIDMutation):
+class SaySomethingAsync(ClientIDMutation):
     class Input:
         what = String()
 
     phrase = String()
 
     @staticmethod
-    def mutate_and_get_payload(self, info, what, client_mutation_id=None):
-        return Promise.resolve(SaySomething(phrase=str(what)))
+    async def mutate_and_get_payload(self, info, what, client_mutation_id=None):
+        return SaySomething(phrase=str(what))
 
 
 # MyEdge = MyNode.Connection.Edge
@@ -97,7 +96,7 @@ class RootQuery(ObjectType):
 class Mutation(ObjectType):
     say = SaySomething.Field()
     say_fixed = SaySomethingFixed.Field()
-    say_promise = SaySomethingPromise.Field()
+    say_async = SaySomethingAsync.Field()
     other = OtherMutation.Field()
 
 
@@ -105,7 +104,7 @@ schema = Schema(query=RootQuery, mutation=Mutation)
 
 
 def test_no_mutate_and_get_payload():
-    with pytest.raises(AssertionError) as excinfo:
+    with raises(AssertionError) as excinfo:
 
         class MyMutation(ClientIDMutation):
             pass
@@ -185,12 +184,13 @@ def test_node_query_fixed():
     )
 
 
-def test_node_query_promise():
-    executed = schema.execute(
-        'mutation a { sayPromise(input: {what:"hello", clientMutationId:"1"}) { phrase } }'
+@mark.asyncio
+async def test_node_query_async():
+    executed = await schema.execute_async(
+        'mutation a { sayAsync(input: {what:"hello", clientMutationId:"1"}) { phrase } }'
     )
     assert not executed.errors
-    assert executed.data == {"sayPromise": {"phrase": "hello"}}
+    assert executed.data == {"sayAsync": {"phrase": "hello"}}
 
 
 def test_edge_query():
diff --git a/graphene/types/schema.py b/graphene/types/schema.py
index be40bd6a..1d84dea0 100644
--- a/graphene/types/schema.py
+++ b/graphene/types/schema.py
@@ -454,13 +454,13 @@ class Schema:
     def lazy(self, _type):
         return lambda: self.get_type(_type)
 
-    async def async_execute(self, *args, **kwargs):
-        return graphql(self.graphql_schema, *args, **normalize_execute_kwargs(kwargs))
+    async def execute_async(self, *args, **kwargs):
+        kwargs = normalize_execute_kwargs(kwargs)
+        return await graphql(self.graphql_schema, *args, **kwargs)
 
     def execute(self, *args, **kwargs):
-        return graphql_sync(
-            self.graphql_schema, *args, **normalize_execute_kwargs(kwargs)
-        )
+        kwargs = normalize_execute_kwargs(kwargs)
+        return graphql_sync(self.graphql_schema, *args, **kwargs)
 
     def introspect(self):
         introspection = self.execute(introspection_query)
diff --git a/tests_asyncio/test_relay_mutation.py b/tests_asyncio/test_relay_mutation.py
index 42ea5fc7..f63ae193 100644
--- a/tests_asyncio/test_relay_mutation.py
+++ b/tests_asyncio/test_relay_mutation.py
@@ -1,5 +1,4 @@
-import pytest
-from graphql.execution.executors.asyncio import AsyncioExecutor
+from pytest import mark
 
 from graphene.types import ID, Field, ObjectType, Schema
 from graphene.types.scalars import String
@@ -64,23 +63,19 @@ class Mutation(ObjectType):
 schema = Schema(query=RootQuery, mutation=Mutation)
 
 
-@pytest.mark.asyncio
+@mark.asyncio
 async def test_node_query_promise():
-    executed = await schema.execute(
+    executed = await schema.execute_async(
         'mutation a { sayPromise(input: {what:"hello", clientMutationId:"1"}) { phrase } }',
-        executor=AsyncioExecutor(),
-        return_promise=True,
     )
     assert not executed.errors
     assert executed.data == {"sayPromise": {"phrase": "hello"}}
 
 
-@pytest.mark.asyncio
+@mark.asyncio
 async def test_edge_query():
-    executed = await schema.execute(
+    executed = await schema.execute_async(
         'mutation a { other(input: {clientMutationId:"1"}) { clientMutationId, myNodeEdge { cursor node { name }} } }',
-        executor=AsyncioExecutor(),
-        return_promise=True,
     )
     assert not executed.errors
     assert dict(executed.data) == {