Begin by exporting the data from Smartsheets in a format that can be easily processed. Smartsheets allows you to export data to CSV or Excel formats. Simply open your Smartsheet, go to 'File', select 'Export', and choose either 'Export to Excel' or 'Export to CSV'. Save the file to a local directory on your computer.

Review and clean the exported data file. Open the CSV or Excel file and ensure that the data is consistent and free of errors. Check for any missing values, incorrect data types, or duplicates that might cause issues during the import process into Apache Iceberg.

Ensure that you have an Apache Iceberg environment ready for use. Install Apache Iceberg on your local system or a server. You will also need a compatible compute engine like Apache Spark or Apache Flink, as Iceberg works with these engines to process data. Follow the official installation guides for Apache Iceberg and your chosen compute engine.

Convert the cleaned CSV or Excel data into a format that Apache Iceberg supports, such as Parquet or ORC. You can use Python with libraries like Pandas and PyArrow, or use Spark directly to read the CSV and write it out as a Parquet file. For example, in Spark, you can use the following commands:
```python
from pyspark.sql import SparkSession

spark = SparkSession.builder.appName("ConvertCSVToParquet").getOrCreate()
df = spark.read.csv("path_to_your_csv_file.csv", header=True, inferSchema=True)
df.write.parquet("path_to_parquet_file.parquet")
```

Within your Iceberg environment, create a new table to store the data. This can be done using SQL commands in your compute engine. For example, in Spark SQL you might run:
```sql
CREATE TABLE my_iceberg_table (
column1_name column1_type,
column2_name column2_type,
...
)
USING iceberg
LOCATION 'path_to_table_location';
```

With the Parquet file ready, load this data into your Iceberg table. Again, using Spark, you can load the data into the table with a command such as:
```python
df = spark.read.parquet("path_to_parquet_file.parquet")
df.write.format("iceberg").mode("append").save("path_to_table_location")
```

Finally, verify that the data has been successfully loaded into the Iceberg table. Use SQL queries in your compute engine to check the data:
```sql
SELECT * FROM my_iceberg_table LIMIT 10;
```
Ensure that the data appears as expected and that there are no discrepancies. Perform any necessary checks to confirm data integrity and completeness.

By following these steps, you can efficiently move data from Smartsheets to Apache Iceberg without relying on third-party connectors or integrations.